A channel coding problem with cost constraint for general channels is considered. Verdú and Han derived ϵ-capacity for general channels. Following the same lines of its proof, we can also derive ϵ-capacity with cost constraint. In this paper, we derive a formula for ϵ-capacity with cost constraint allowing overrun. In order to prove this theorem, a new variation of Feinstein's lemma is applied to select codewords satisfying cost constraint and codewords not satisfying cost constraint.

In the source coding problem with cost constraint, a cost function is defined over the code alphabet. This can be regarded as a noiseless channel coding problem with cost constraint. In this case, we will not distinguish between the input alphabet and the output alphabet of the channel. However, we must distinguish them for a noisy channel. In the channel coding problem with cost constraint so far, the cost function is defined over the input alphabet of the noisy channel. In this paper, we define the cost function over the output alphabet of the channel. And, the cost is paid only after the received word is observed. Note that the cost is a random variable even if the codeword is fixed. We show the channel capacity with cost constraint defined over the output alphabet. Moreover, we generalize it to tolerate some decoding error and some cost overrun. Finally, we show that the cost constraint can be described on a subset of arbitrary set which may have no structure.

The wiretap channel is now a fundamental model for information-theoretic security. After introduced by Wyner, Csiszár and Körner have generalized this model by adding an auxiliary random variable. Recently, Han, Endo and Sasaki have derived the exponents to evaluate the performance of wiretap channels with cost constraints on input variable plus such an auxiliary random variable. Although the constraints on two variables were expected to provide larger-valued (or tighter) exponents, some non-trivial theoretical problems had been left open. In this paper, we investigate these open problems, especially concerning the concavity property of the exponents. Furthermore, we compare the exponents derived by Han et al. with the counterparts derived by Gallager to reveal that the former approach has a significantly wider applicability in contrast with the latter one.